Trigonometric Graphs and Equations

This revision note aims to aid in mastering trigonometric graphs and solving trigonometric equations, vital components of advanced mathematics.

1. Introduction to Basic Trigonometric Functions

Key Functions

• Sinusoidal Functions: These primary functions are crucial in understanding oscillatory behaviours.
  • Sine Function (sin x): Describes oscillations, exhibiting wave-like patterns.
  • Cosine Function (cos x): Also periodic and, together with sine, defines circular movement.
  • Tangent Function (tan x): Expressed as the ratio $\frac{\sin x}{\cos x}$, it highlights periodic vertical asymptotes.

Key Definitions

Domains and Ranges

• Sine and Cosine:

  • Domain: All real numbers, ($-\infty, \infty$).
  • Range: [$-1, 1$].
  • Graph Features:
    • They repeat every $2\pi$, demonstrating symmetry and periodicity.

• Tangent:

  • Domain: Excludes points $\frac{\pi}{2} + n\pi$ where $n$ is an integer.
  • Range: All real numbers ($-\infty, \infty$).
  • Graph Features:
    • Asymptotes and a period of $\pi$.

2. Understanding Properties and Visualisation

Fundamental Properties

• Periodicity:

  • $\sin x$ and $\cos x$ both display a period of $2\pi$.
  • $\tan x$ displays a period of $\pi$.

• Symmetry:

  • Odd Functions:
    • $\sin x$ is odd: $\sin(-x) = -\sin x$.
    • $\tan x$ is odd: $\tan(-x) = -\tan x$.
  • Even Function:
    • $\cos x$ is even: $\cos(-x) = \cos x$.

• Amplitude:

  • For $\sin$ and $\cos$: The maximum height is 1.

Visual Aids and Graphs

• Utilise graphs to illustrate periodicity, symmetry, and key features like maximum/minimum values, utilising visual aids when feasible.

3. Introduction to Transformations

Transformation Equation: To modify trigonometric functions, use:

$y = kf(a(x + b)) + c$

Components and Their Effects

• Amplitude ($k$): Adjusts the wave's height.
• Period ($a$): Affects the frequency.
• Phase Shift ($b$): Causes horizontal displacement.
• Vertical Shift ($c$): Introduces vertical changes.

Practical Transformation Examples

• Amplitude Change: Example $y = 3\sin(x)$ increases the amplitude to 3.

• Period Adjustment: $y = \cos(2x)$ results in $T = \pi$, compressing the cycle.

• Phase Shift Alteration: In $y = \sin(x + \frac{\pi}{4})$, the graph shifts left by $\frac{\pi}{4}$.

• Vertical Shift: For $y = \sin(x) + 2$, the graph moves upwards by 2 units.

• Visualisation and Diagram Usage

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Common Misunderstandings

• Often, students confuse directions of phase shifts—precision is crucial to avoid tracing errors.
• Graphing tools are invaluable in enhancing visual comprehension.

4. Trigonometric Equations and Solutions

Key Strategies

• Transformation Strategy: Simplifies equations for easier handling.
• Horizontal and Vertical Shifts: Key in adjusting function positions.

Practical Approach Example

• Consider $2\cos(x) - 1 = 0$:

  • Isolate: $\cos(x) = \frac{1}{2}$
  • Solutions are found as $\frac{\pi}{3}$ and $\frac{5\pi}{3}$ for the interval $[0, 2\pi)$.

• Further Illustration

  • Utilise sketches and verify visually with graphs and detailed algebra.

Conclusion

Apply this understanding to execute further examples, reinforcing transformations and solutions in trigonometric functions. Engage with software to broaden insights, preparing effectively for examinations.